A360772 List of distinct numbers that are powers of odd-indexed Fibonacci numbers or even powers of nonzero even-indexed Fibonacci numbers.

1, 2, 4, 5, 8, 9, 13, 16, 25, 32, 34, 64, 81, 89, 125, 128, 169, 233, 256, 441, 512, 610, 625, 729, 1024, 1156, 1597, 2048, 2197, 3025, 3125, 4096, 4181, 6561, 7921, 8192, 10946, 15625, 16384, 20736, 28561, 28657, 32768, 39304, 54289, 59049, 65536, 75025, 78125

Offset: 1

Author

Amiram Eldar, Feb 20 2023

Keywords

• nonn

Comments

Ohtsuka's (2023) problem does not include 1, and includes the even powers of 8 twice (once as powers of Fibonacci(6) = 8 and once as powers of Fibonacci(3) = 2). The sum of reciprocals in this case is (61 - 15*sqrt(5))/18.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000
• Hideyuki Ohtsuka, Problem B-1321, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 61, No. 1 (2023), p. 84.

Crossrefs

Cf. A000045, A001519, A001906, A103323, A105317, A235383, A254719.

Programs

Mathematica

seq[max_] := Module[{s = {1}, k = 3, f, d}, While[(f = Fibonacci[k]) <= max, If[k != 6, d = 2 - Mod[k, 2]; s = Join[s, f^Range[d, Floor[Log[f, max]], d]]]; k++]; Sort[s]]; seq[10^5]

Formula

Sum_{n>=1} 1/a(n) = 551/126 - 5*sqrt(5)/6.